Two-point Correlation Functions in Inhomogeneous and Anisotropic Cosmologies
PPrepared for submission to JCAP
Two-point Correlation Functions inInhomogeneous and AnisotropicCosmologies
Oton H. Marcori, a Thiago S. Pereira a a Departamento de Física, Universidade Estadual de Londrina, 86057-970, Londrina PR,BrazilE-mail: [email protected], [email protected]
Abstract.
Two-point correlation functions are ubiquitous tools of modern cosmology, appear-ing in disparate topics ranging from cosmological inflation to late-time astrophysics. Whenthe background spacetime is maximally symmetric, invariance arguments can be used to fixthe functional dependence of this function as the invariant distance between any two points.In this paper we introduce a novel formalism which fixes this functional dependence directlyfrom the isometries of the background metric, thus allowing one to quickly assess the overallfeatures of Gaussian correlators without resorting to the full machinery of perturbation theory.As an application we construct the CMB temperature correlation function in one inhomo-geneous (namely, an off-center LTB model) and two spatially flat and anisotropic (Bianchi)universes, and derive their covariance matrices in the limit of almost Friedmannian symmetry.We show how the method can be extended to arbitrary N -point correlation functions andillustrate its use by constructing three-point correlation functions in some simple geometries. Keywords:
Correlation functions, spacetime symmetries, CMB a r X i v : . [ g r- q c ] F e b ontents A.1 Power spectrum and Hankel transform 19A.2 Covariance matrix 19A.3 Gaunt coefficients 20A.4 Bipolar power spectrum 20
A central assumption of the standard cosmological model is that the universe we observe isa fair sample of an (hypothetical) ensemble of universes. This hypothesis has far reachingconsequences, but it also brings along a whole statistical framework from which cosmologicalobservables are to be computed. It follows in particular that cosmological parameters are notdeduced directly from physical fields – which in this framework are viewed as one realizationof random variables – but rather from their statistical moments, such as the one, two, andhigher N -point correlation functions. When using perturbation theory to describe the clumpyuniverse, the one-point function is usually defined to be zero, since one is actually interestedin the fluctuations of physical fields around their mean values. Thus, the first non-trivialstatistical moment is the two-point (or Gaussian) correlation function.Two-point functions are ubiquitous tools in modern physics. In field theory they aredisguised as Green’s functions (or the propagator), whereas in general relativity they could besimply a distance function or a bitensor [1, 2] – just to mention a few examples. In cosmology,two-point correlation functions are a cornerstone of the standard Λ CDM model. Once it arisesas the quantization of a free field in the early inflationary universe [3–5] (see ref. [6] for anup-to-date review), it propagates to virtually all cosmological and astrophysical computationsone might be interested in – most popularly in its Fourier (i.e., the power spectrum) version.The same reasoning holds for higher-order correlation functions in connection with “Beyond- Λ CDM” approaches [7, 8]. Therefore, knowledge of the functional dependence of the two-point– 1 –orrelation function (2pcf) is crucial, since it alone can tell a lot about the statistical propertiesof cosmological observables, potentially allowing one to disentangle cosmological signals fromsystematical effects in real data.There are essentially two independent routes to find the functional dependence of the2pcf in cosmology. In the first, one uses heuristic symmetry arguments (or its lack thereof) tofix this functional dependence. This idea has been successfully applied in cosmology, mainly inconnection with CMB physics, in refs. [9–14]. However straightforward, the phenomenologicalquality of this approach prevents one to link the resulting 2pcf to the statistics of a field ina well-defined background geometry. Alternatively, one can deploy the full machinery ofperturbation theory in the desired spacetime. After dealing with known issues of gaugeinvariance and mode decomposition, the full set of Einstein equations can be solved and thestatistics of the 2pcf can be computed [15–19]. This option is clearly more expensive, but iscertain to lead to statistics with known spacetime symmetries.These considerations lead us to ask whether one can systematically find the functionalproperties of correlation functions given the spacetime symmetries, and without the need toresort to expensive computations involving perturbation theory. In fact, when metric andfluid perturbations are small, they can be seen as external fields evolving over a fixed back-ground, regardless of their dynamics. By expanding such fields in an appropriate set of basiseigenfunctions, one ensures that their statistical properties will inherit the symmetries of thebackground metric. Thus, in a Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime,for example, the 2pcf of a random field can only depend on the invariant distance betweenthe two points, since this is the only combination allowed by the symmetries of the FLRWmetric. Analogous ideas were explored in refs. [20, 21], where the conformal invariance ofthe de Sitter spacetime has been used to find the shape of two- and three-point correlationfunctions in dark-energy dominated universes.In this work we systematically develop the idea of using the symmetries of the back-ground metric to fix the functional form of the 2pcf. Starting from the definition of a two-pointfunction in a general manifold, we show in §2 that the imposition of isometric invariance onthe 2pcf leads to a set of coupled first order partial differential equations which can be solvedby means of well known techniques – but most easily through the method of characteristiccurves [22] – to fix the functional form of the 2pcf. We illustrate the method in §2.1, wherewe show how it correctly recovers the 2pcf in spatially flat FLRW spacetimes. We end thissection by constructing in §2.2 a formal solution to the aforementioned set of differentialequations which holds for any spacetime having at least one Killing vector. In §3 we applythe formalism to obtain the 2pcf in two different classes of spacetimes. First, we consider thecase of an off-center inhomogeneous but spherically symmetric spacetime. Then we show howthe 2pcf will appear in a class of homogeneous but spatially flat anisotropic geometries of theBianchi family. In both cases we derive the CMB temperature covariance matrix in the limitof almost Friedmannian symmetry, and comment on their multipolar signatures. In §4 weshow how the method can be easily generalized to include any N -point correlation function.We conclude with some final remarks in §5. We start with an informal description of what is meant by a two-point correlation functionin spacetime. For a rigorous and mathematically complete description of two-point functionsin Riemannian spaces, see ref. [23]. – 2 – two-point function f on a manifold M is simply a real valued function of a pair ofpoints ( p, q ) ∈ M × M . Known examples in physics are Green’s functions, the geodesicdistance between two points or Synge’s world function [2]. Here we shall be mainly interestedin correlation functions, so we also demand f to be symmetric f ( p, q ) = f ( q, p ) , (2.1)since correlation is clearly a pairwise concept. In most interesting situations in cosmologyone is dealing with the correlation of random variables in spacetimes with some symmetries.Whenever these variables can be viewed as external fields over a fixed background, theirstatistical properties will inherit the symmetries of the underlying space. We would thus liketo define an invariant correlation function with respect to these symmetries. Suppose that M possesses an isometry represented by a one-parameter family of diffeomorphisms φ τ thatmaps any point p ∈ M to the point φ τ ( p ) ∈ M such that φ ( p ) = p . Clearly, f will beinvariant under this symmetry if f ( p, q ) = f ( φ τ ( p ) , φ τ ( q )) . (2.2)In practice, though, one is always working in a specific coordinate patch. Suppose that ψ isa chart on an open interval of M and define f ◦ ψ − = f (cid:0) ψ − ( x µ ) , ψ − ( x µ ) (cid:1) ≡ ξ ( x µ , x µ ) . (2.3)Therefore, the components of the curve φ τ in the coordinate system defined by ψ are ( ψ ◦ φ τ ) µ | p = x µ ( τ ) , and ( ψ ◦ φ τ ) µ | q = x µ ( τ ) . (2.4)Locally, condition (2.2) then reads ξ ( x µ , x µ ) = ξ ( x µ ( τ ) , x µ ( τ )) (2.5)which, for infinitesimal τ , is equivalent to K ( ξ ) = 0 , (2.6)where K = d/dτ is a Killing vector, i.e., the vector tangent to the curves generated by φ τ .This condition is nothing more than K µ ∂ µ ξ | p + K µ ∂ µ ξ | q = 0 , K µ = dx µ dτ . (2.7)Notice that in deriving this formula we are implicitly assuming that both p and q are coveredby the same coordinate system. Since in general M can have several independent isometries,we generalize the above result to the set of equations K µ a ∂ µ ξ | p + K µ a ∂ µ ξ | q = 0 , a ∈ Isom ( M ) (2.8)where Isom ( M ) is the set of all isometries of M . As we will see, this set of equations fullydetermine the functional dependence of the 2pcf.– 3 – .1 Example: spatially flat FLRW universe Equations (2.8) form the core of our formalism. They will lead to a set of coupled firstorder partial differential equations which can be implicitly solved by means of the methodof characteristics curves [22]. In order to illustrate the method let us consider a two-pointfunction in a spatially flat FLRW universe; it could be, for example, the ensemble averageof the gravitational potential at two points on the same time slice. Since FLRW universesare maximally symmetric expanding manifolds they possess six independent Killing vectors:three of translation ( T i ) and three of rotation ( R i ). In Cartesian coordinates these vectorsread T i = ∂ i , R i = (cid:15) ijk x j ∂ k . (2.9)The two-point function depend on six variables: ξ = ξ ( x , y , . . . , z ) . In practice it is easierto work with ( ± ) -coordinates defined as x ± = x ± x , y ± = y ± y , z ± = z ± z , (2.10)so that ξ = ξ ( x − , . . . , z + ) . Let us start with the vector T x . In Cartesian coordinates we havethat T x = T µx ∂ µ = (1 , , , which implies T µx = δ µx . Thus, for this KV, equations (2.8) give: ∂ξ∂x + = 0 . (2.11)Clearly, ξ cannot depend on x + . Since this conclusion will not be straightforward in general,let us illustrate how it follows from the method of characteristics. Let τ be the parameteralong the integral curves (i.e., the isometry) of T x . Thus, by definition the tangent vector tothis isometry is T x = d/dτ , and we have by virtue of eq. (2.6) that T x ( ξ ) = ˙ x − ∂ξ∂x − + ˙ x + ∂ξ∂x + + · · · + ˙ z + ∂ξ∂z + = 0 (2.12)where a dot means a (partial) derivative with respect to τ . Comparing (2.11) and (2.12) wesee that all coordinates are constant along τ except for x + . Therefore ξ cannot depend onit. A similar procedure using T y and T z tell us that ξ cannot depend on either y + or z + , sothat ξ = ξ ( x − , y − , z − ) . Consider next the vector R z = d/dρ . Using R z = ( − y, x, on (2.8)we find x − ∂ξ∂y − − y − ∂ξ∂x − = 0 . (2.13)On the other hand, we also have that dξdρ = ˙ x − ∂ξ∂x − + ˙ y − ∂ξ∂y − + ˙ z − ∂ξ∂z − = 0 (2.14)where a dot now stands for ∂/∂ρ . By comparing the last two equations we find ˙ x − = − y − , ˙ y − = x − , ˙ z − = 0 . (2.15)The first pair of equations can be easily decoupled, giving (after an arbitrary choice of phase) x − = A cos ρ , y − = A sin ρ , (2.16)– 4 –here A is not necessarily a constant, since it can depend on the parameters of other isome-tries. The most general and ρ -independent combination of x − and y − is x − + y − = A , sothat ξ = ξ (cid:0) x − + y − , z − (cid:1) . Moving on we now consider the vector R y . By the same reasoningwe find z − ∂ξ∂x − − x − ∂ξ∂z − = 0 . (2.17)However we note that in virtue of (2.16) x − and y − are not independent anymore. We thusdefine u − ≡ x − + y − so that equation above becomes z − x − u − ∂ξ∂u − − x − ∂ξ∂z − = 0 . (2.18)If we now use R y = d/dr and expand dξ/dr as a total derivative we find by comparison that u − ˙ u − = z − x − , ˙ z − = − x − (2.19)which is easily solved by u − = B sin r , z − = B cos r , (2.20)with B constant. Thus, a ρr -independent combination of variables is x − + y − + z − , whichincidentally tells us that A = B sin r . Introducing the notation r , = ( x , , y , , z , ) wefinally conclude that ξ ( | r − r | ) ≡ ξ F L ( r − ) (2.21)gives a solution to (2.8) up to an overall power of the argument. While we are on this topic,let us further remark that for CMB large angle perturbations the above solution becomes ξ F L = ξ F L (cid:16) ∆ η (cid:112) − γ (cid:17) = (cid:88) (cid:96) (cid:96) + 14 π C (cid:96) P (cid:96) (cos γ ) , (2.22)where γ is the angle between r and r and ∆ η is the distance to the last scattering surface.Finally, note that there is no need to impose the condition R x ( ξ ) = 0 since it is automaticallyensured by the algebra of KVs : R x ( ξ ) = − [ R y , R z ] ( ξ ) = 0 .The method above also works for timelike isometries. For example, in Minkowski space-time one has, additionally to the vectors above, three boost KVs and one time-translation KV.One can easily check that the same procedure will give ξ = ξ (∆ s ) , where ∆ s = ( η µν x µ − x ν − ) / .In the next section we will show that eq. (2.8) can be formally solved for any spacetime havingat least one isometry. In proving this solution we will thus arrive at an independent formula-tion of the problem which in some cases (most notably in Bianchi spacetimes) is simpler thanthe above examples, and which also takes care of the time dependence. As we have seen, the condition for a two-point correlation function – or any two-point function– to be invariant with respect to an isometry represented by the Killing vector K , whoseintegral curves are measured by a parameter τ , is dξdτ = 0 . (2.23) The variable z − is already ρ -independent, so it does not enter into this combination. Physically, this steems from the fact that the constancy of two components of the angular momentumvector implies the constancy of the third. – 5 – igure 1 . Schematic representation of two points on the manifold connected by the vector e = d/ds and dragged by the Killing vector K = d/dτ . Suppose now that e is a vector field commuting with K , that is £ K e µ = [ K , e ] µ = 0 . (2.24)Then e is a vector connecting two close points on different curves generated by K . Moreoverthe quantity u = g µν e µ e ν is obviously constant along these isometries, since £ K u = ( £ K g µν ) e µ e ν + 2 g µν ( £ K e µ ) e ν = 0 (2.25)where the first term is zero since K is a Killing vector. This suggests that eq. (2.23) will besolved for any function of u . That is ξ = ξ (cid:18) ˆ s s (cid:112) g µν e µ e ν ds (cid:19) , (2.26)with e µ = ∂x µ /∂s , is a solution of (2.23), since dξ/dτ = ( dξ/du )( du/dτ ) = 0 . We have thusfound a general solution to eq. (2.23). Since in general we will have more than one Killingvector, the 2pcf we will have more then one argument, provided that we can find a set ofindependent vectors { e i } commuting with the vectors { K i } . When this is the case ξ = ξ (cid:18) ˆ r r (cid:113) g µν e µ e ν d r, ˆ s s (cid:113) g µν e µ e ν d s, . . . (cid:19) (2.27)will be a solution to (2.8). This solution is particularly suited to the construction of 2pcfin homogeneous and anisotropic Bianchi geometries where the basis { e i } can always be con-structed from the conditions [ K i , e j ] = 0 . In these cases the metric can be written as [24, 25] d s = − d τ ⊗ d τ + e α ( τ ) (cid:16) e β ( τ ) (cid:17) ij e i ⊗ e j . (2.28)Here the vectors { e i } are the duals to { e i } , (cid:0) e β (cid:1) ij is a symmetric and traceless × matrixwhose eigenvalues are the directional scale factors, and e α is the geometrically averaged scalefactor. For these vectors we have (no sum over i ) g µν e µi e νj = e α ( τ ) e β ii ( τ ) δ ij . (2.29)– 6 –his implies in particular that ˆ r r (cid:113) g µν e µ e ν d r = e α ( τ ) e β ( τ ) ( r − r ) (2.30)with similar expressions for the other arguments. We have thus arrived at a formal expressionfor the 2pcf which is valid in any Bianchi spacetime ξ = ξ (cid:16) e α ( τ ) e β ( τ ) ( r − r ) , e α ( τ ) e β ( τ ) ( s − s ) , e α ( τ ) e β ( τ ) ( t − t ) (cid:17) . (2.31)To convert this function to one valid in an specific coordinate system one have to find theparametric curves of the vectors e i in the desired coordinates and invert these relationsto obtain the parameters as a function of the coordinates. Of course, the success of thisprocedure depends on the coordinate system chosen. We will illustrate this method withexplicit examples in next section, where we find ξ for the geometries of Bianchi I and VII universes. We are now in position to put the above formalism to practical use. We start in §3.1 withthe example of an inhomogeneous universe with an off-center special point around which it isspherically symmetric. This could be seen as an off-center LTB spacetime, though in realityany spherically symmetric solution with a privileged point will lead to the same answer. Thenin §3.2 we consider two anisotropic spacetimes with spatially flat spatial sections – namely,the models of Bianchi I and VII . We then derive the Friedmannian limit of the 2pcf withfirst order corrections in both cases, and connect the result with the temperature covariancematrix of CMB fluctuations in §3.3. The 2pcf in an universe with a special point was studied from a phenomenological standpointin ref. [11]. More recently, the effect of an off-center spherically symmetric void on thefrequency and polarization of CMB photons was investigated by the authors of ref. [26].Here we shall model an off-center spherically symmetric universe by its Killing vectors. Let w = ( a, b, c ) represent the spatial coordinates of this point with respect to our frame. Thenthe only isometries are rotations about w . These are represented by the following KVs: R i = (cid:15) ijk (cid:0) x j − w j (cid:1) ∂ k . (3.1)Let us start with rotations around the z -axis. Applying R z = ( − y + b, x − a, to eq. (2.8)leads to ( x + − a ) ∂ξ∂y + + x − ∂ξ∂y − − ( y + − b ) ∂ξ∂x + − y − ∂ξ∂x − = 0 . (3.2)Let ρ be the parameter along the integral curves of R z , such that R z = d/dρ . Comparingthe above with dξ/dρ = 0 gives ˙ x + = − y + + 2 b , ˙ x − = − y − , ˙ y + = x + − a , ˙ y − = x − , ˙ z ± = 0 . (3.3)– 7 –fter decoupling and solving these equations we find that the combinations u − ≡ x − + y − and v + ≡ ( x + − a ) + ( y + − b ) are constants with respect to ρ , so that ξ = ξ ( u − , v + , z − , z + ) .We next consider R y = ( z, , − x ) and change variables from ( x − , x + ) to ( u − , v + ) . This gives z − x − u − ∂ξ∂u − + ( z + − c ) ( x + − a ) v + ∂ξ∂v + − ( x + − a ) ∂ξ∂z + − x − ∂ξ∂z − = 0 . (3.4)We now compare this to dξ/dr = 0 , where r is such that R y = d/dr . This gives u − ˙ u − = z − x − , v + ˙ v + = ( z + − c ) ( x + − a ) , ˙ z + = − x + + 2 a , ˙ z − = − x − . (3.5)Combining the last equation with the first and the third with the second we find two constantcombinations of variables: u − + z − and v +( z + − c ) . Thus, ξ = ξ (cid:16) u − + z − , v + ( z + − c ) (cid:17) .After a little algebra on the second argument, the final solution can be written as ξ w = ξ w (cid:16) | r − r | , | r − w | + | r − w | + 2 ( r − w ) · ( r − w ) (cid:17) . (3.6)This result is compatible with the one found heuristically by the authors of ref. [11]. Notehowever that the above solution is more restrictive than theirs, since here we can obviouslywrite ξ w = ξ w ( | r − r | , | r + r − w | ) (3.7)whereas in [11] this is not possible .Before continuing we would like to make two comments about solution (3.7). First,the casual reader could be worried that the above solution does not seem to recover (2.21)when w = 0 . This happens because (3.7) is only invariant under rotations, whereas (2.21)also obeys translation symmetry. If the universe is homogeneous then w = 0 and we canfurther impose translation invariance through the condition T r ( ξ ) = ∇ r ξ + ∇ r ξ = 0 , thuseliminating the dependence on | r + r | . As a corollary of this result we find that the 2pcf ina inhomogeneous but spherically symmetric universe about its origin – such as in Lemaître-Tolman-Bondi (LTB) universes – is ξ = ξ ( | r − r | , | r + r | ) (3.8)which is functionally equivalent to ξ ( r , r , ˆ r · ˆ r ) , since the only angle entering eq. (3.8) isthat between r and r . The second remark is that, as stressed in [11], the 2pcf (3.7) possessa global shift symmetry of the form r , → r , + a , w → w + a (3.9)for any vector a . This corresponds to the freedom in placing a special point in an otherwisehomogeneous universe, which is only defined up to a global translation. We will come backto this issue in §4.Equation (3.7) concludes our task of finding the 2pcf in an universe with a special point.The question of whether we actually live close to an off-center spherically symmetric universecan be tested by measuring off-diagonal terms in the covariance matrix of CMB temperaturefluctuations. Since we know that our universe is very close to FLRW, we can test this This difference does not affect the conclusions found by those authors, since most of their results areactually extracted from an ansatz of the power spectrum, and not from ξ . – 8 – igure 2 . Schematic representation of a universe with a special point Q at a distance w from ourposition at P . In such universe the correlation between two photons coming from positions A and B can only depend on | r − r | and | r + r − w | . hypothesis by deriving the FLRW limit of eq. (3.7), including leading order corrections. Todo that we first introduce the variables r ± = r ± r . (3.10)Then we note that the desired limit of (3.7) involves two independent expansions, namely,one in | w | and another one in powers of r + = | r + | . Let us start with the former. Assuming | w | (cid:28) we have | r + − w | = r + − n + · w + O ( | w | ) . (3.11)Thus ξ w ≈ ξ w ( r − , r + − n + · w )= ξ ( r − , r + ) − ∂ξ ( r − , r + ) ∂r + ˆ n + · w + · · · . (3.12)Next we assume that ξ ( r − , r + ) varies weakly with r + and write ξ ( r − , r + ) = ξ ( r − ,
0) + ∂ξ ( r − , ∂r + r + + · · · . (3.13)It is important to note that we are not treating r + as a small parameter. Indeed, this willhardly be the case, since for coincident points on the CMB sphere we have r + = 2∆ η , whichis not assumed as small. On the other hand, the assumption that ξ varies weakly with r + implies that its translational invariance is only slightly broken. That is T r ( ξ ( r − , r + )) = ∇ r + ξ ( r − , r + ) = ∂ξ ( r − , ∂r + ˆ n + (cid:28) . (3.14)Since ξ ( r − ,
0) = ξ F L ( r − ) , we finally find that ξ w = ξ F L ( r − ) + ∂ξ ( r − , ∂r + ( r + − n + · w ) , (3.15)– 9 –hich is the desired result .In order to extract the amplitude of the leading corrections one still needs the specificshape of the function ξ ( r − , r + ) , which at this point can only be fixed from first physicalprinciples [16, 26, 27]. Note however that, as far as the angular dependence is concerned,there is no new information in ξ ( r − , r + ) as compared to (2.22), since its angular dependencealso comes from the angle between r and r . This means that the middle term in (3.15) willnot induce off-diagonal correlations in the CMB covariance matrix, although it will surelyalter the amplitude of the isotropic temperature spectrum, i.e., the C (cid:96) s. On the other hand,the last term will induce a dipole coming from the angle between ˆ n + and w . We show in §3.3how these multipolar coefficients can be directly linked to the temperature covariance matrix. In order to derive the 2pcf in Bianchi universes we start from the general solution (2.31),which we have already proven to solve (2.8). One can check that the same results followinstead from the direct application of eq. (2.8), up to the dependence on the directional scalefactors. We will here focus on two spatially flat anisotropic solutions and postpone a completeanalysis with other Bianchi metrics to a future work.
We start with the simple Bianchi-I metric, which admits three translational KVs: T x = ∂ x , T y = ∂ y , T z = ∂ z . (3.16)The set of triad { e i } vectors which are invariant under the action of these isometries are[28, 29] e = (1 , , , e = (0 , , , e = (0 , , . (3.17)Let us solve for the integral curves of the first vector. Putting e i = dx i /dr we find that x = r , y = z = constant. Thus, we can invert the relation between the parameter and thecoordinates to find r − r = x − x = x − . (3.18)Solving for e and e leads to s − s = y − and t − t = z − . Plugging these results back into(2.31) then gives ξ I = ξ I (cid:16) e α ( τ ) e β ( τ ) x − , e α ( τ ) e β ( τ ) y − , e α ( τ ) e β ( τ ) z − (cid:17) , (3.19)which is the desired solution.Observational evidences tell us that our universe is very close to isotropic [30–32]. Wecan thus obtain the FLRW limit of ξ I by Taylor expanding this function around β ii = 0 . Wefind ξ I = ξ I ( x − , y − , z − ) + (cid:20) β x − ∂∂x − + β y − ∂∂y − + β z − ∂∂z − (cid:21) ξ I ( x − , y − , z − ) + · · · (3.20) Rigorously speaking (3.15) should also include second order terms since ∂ξ /∂r + ( n + · w ) is formed fromthe product of two small quantities. However, second order corrections from (3.11) will multiply ∂ξ /∂r + in(3.12), producing a third order term. The only remaining second order term is a correction to (3.13), but thisdoes not induce any angular dependence on ξ w . – 10 – igure 3 . Integral curves of invariant vectors for Bianchi VII spacetime. These lead to quadrupolecorrections in the covariance matrix, as well as a change in the low- (cid:96) isotropic C (cid:96) s. See the text fordetails. where we have omitted the the functional dependence on α for simplicity. In order to proceed,we note that the failure of the above expression to be rotationally invariant is proportionalto β ii , in the sense that full rotational isotropy should be exactly recovered if β ii = 0 . In fact,by applying, say, R z to the above expression we find R z ( ξ I ) = R z ( ξ I ( x − , y − , z − )) + β ( · · · ) + β ( · · · ) + β ( · · · ) (3.21)where the ellipses contain terms like x − ∂ y − (cid:0) x − ∂ x − ξ (cid:1) and so on, but which are not relevant forthis discussion. The important point is that the right hand side is linear in β ii . Therefore, for ξ I to be rotationally invariant at zero-order in β ii , it is necessary that R z ( ξ I ( x − , y − , z − )) = 0 at this order. Repeating the analysis with R y or R x then leads to the isotropic condition ξ I ( x − , y − , z − ) = ξ F L ( r − ) . (3.22)Thus, the FLRW limit of (3.19) including first order anisotropic corrections is ξ I = ξ F L + 1 r − ∂ξ F L ∂r − (cid:2) β x − + β y − + β z − (cid:3) . (3.23)Clearly, this function will induce quadrupolar corrections in the statistics of CMB, as isalready known [15, 29, 33, 34]. Interestingly, though, it does not alter the isotropic spectrum,as we will see in §3.3. The spatial topology of Bianchi-VII solution is R , so that it also has a flat FLRW limitwhen β ii = 0 . The isometries of this space can be seen as two orthogonal displacements inthe xy -plane, and a displacement in the z -axis followed by a rotation in the xy -plane [28, 29].In Cartesian coordinates the three KVs are T x = ∂ x , T y = ∂ y , T z = ∂ z + x∂ y − y∂ x . (3.24) Note that we chose a different orientation of the axis as compared to refs. [28, 29]. – 11 –he set of of invariant triad vectors are [28, 29] e = (cos z, sin z, , e = ( − sin z, cos z, , e = (0 , , . (3.25)Let us consider the first vector with components e i = dx i /dr . Its integral curves are x ( r ) = x + (cos z ) r , y ( r ) = y + (sin z ) r , z ( r ) = z . (3.26)These relations can be easily inverted to give one of the curve segments between the twopoints as a function of their coordinates: r − r = (cid:113) x − + y − . (3.27)A similar computation involving e i = dx i /ds and e i = dx i /dt then gives the other twosegments of the curve s − s = (cid:113) x − + y − , t − t = z − . (3.28)This completes the task of finding the parameters of the integral curves of the vectors { e i } as a function of the coordinates. Inserting the above expressions in (2.31) finally gives ξ V II = ξ V II (cid:18) e α ( τ ) e β ( τ ) (cid:113) x − + y − , e α ( τ ) e β ( τ ) (cid:113) x − + y − , e α ( τ ) e β ( τ ) z − (cid:19) . (3.29)The isotropic limit of ξ V II follows the same discussion of the last section. The only differenceis that, at zero-order in β ii , ξ V II is automatically invariant under R z , as follows from itsisometries. Thus we just require that R x ( ξ V II ) = 0 at zero order, which then gives ξ V II (cid:18)(cid:113) x − + y − , (cid:113) x − + y − , z − (cid:19) = ξ F L ( r − ) . (3.30)Thus ξ V II = ξ F L ( r − ) + 1 r − ∂ξ F L ∂r − (cid:2) ( β + β ) (cid:0) x − + y − (cid:1) + β z − (cid:3) (3.31)completes the desired expansion. Equations (3.7), (3.19), and (3.29), together with their FLRW expansions, are the main resultsof last section. We emphasize that these results are completely general and can be equallyapplied to large scale structures as well as to CMB physics. Here we are interested in thelatter, so that we shall now derive the multipolar expansion of functions (3.15), (3.23), and(3.31) and relate them to the corresponding CMB covariance matrix in the limit of largeangles, i.e., assuming only the Sachs-Wolfe effect.We start by recalling that all the 2pcfs that we are considering have the generic form ξ ( r , r ) = (cid:40) ξ ( r − ) for Bianchi I and VII ,ξ ( r + , w ) for a universe with a special point , (3.32)where ( r + , r − ) were defined in (3.10) and w is the special point introduced in §3.1. Forsimplicity, we have omitted any extra dependence on ( r + , r − ) , since these will not lead to– 12 – able 1 . Non-zero multipolar coefficients of anisotropic (Bianchi-I and Bianchi-VII ) and inhomo-geneous (off-center LTB) 2pcfs considered in this work (respectively eqs. (3.7), (3.19), and (3.29)).The trace-free condition (cid:80) i β ii = 0 was used in all Bianchi solutions. Note that the multipolar coeffi-cients obey the reality condition ξ ∗ (cid:96)m = ( − m ξ (cid:96), − m . Primes in ξ and ξ F L means ∂/∂r + and ∂/∂r − ,respectively. Geometry ξ / √ π ξ ξ ξ Off-center LTB ξ F L + ξ (cid:48) r + − w (cid:112) π ξ (cid:48) ξ F L β (cid:113) π r − ξ (cid:48) F L ( β − β ) (cid:113) π r − ξ (cid:48) F L
Bianchi-VII ξ F L − β r − ξ (cid:48) F L β (cid:113) π r − ξ (cid:48) F L ξ ( r + , w ) on w , thus calling both 2pcfsabove generically as ξ ( r ± ) . This allows us to expand the ξ ( r , r ) collectively as ξ ( r ± ) = (cid:88) (cid:96),m ξ (cid:96)m ( r ± ) Y (cid:96)m (ˆ n ± ) , r ± = r ± ˆ n ± . (3.33)Next we set r ± = r ± (sin θ ± cos φ ± , sin θ ± cos φ ± , cos θ ± ) , w = w ˆ z , (3.34)where θ + = arccos (ˆ n + · ˆ z ) , and extract the coefficients ξ (cid:96)m . Notice that for r + we havedefined the z axis along w . The resulting expressions are collected in Table 1, and can bedirectly related to the CMB temperature covariance matrix, as we now show.At large scales the Sachs-Wolfe effect ( ∆ T /T = Φ / ) gives the main contribution to thetemperature fluctuations. In order to compute the full effect of inhomogeneous or anisotropicgeometries in a real CMB map, gravitational evolution and re-ionization effects should betaken into account. Clearly, such effects will not be provided by our formalism, which isgeometric in nature. On the other hand, we can picture a scenario in which the asymmetriesof the early universe are washed out by inflation, but where quantum fluctuations preservesuch asymmetries on the statistics of the primordial gravitational potential. This is theapproach followed in, e.g., refs. [10, 15, 34]. In this scenario, primordial inhomogeneities andanisotropies are contained in the statistics of CMB, and subsequent evolutionary effects areassumed to be isotropic. Under this assumption the CMB covariance matrix reads (cid:10) a (cid:96) m a ∗ (cid:96) m (cid:11) ± = 19 ˆ d n ˆ d n (cid:104) Φ( r )Φ( r ) (cid:105) ± Y ∗ (cid:96) m (ˆ n ) Y (cid:96) m (ˆ n ) . (3.35)The two-point correlation function in this case is the ensemble average of the gravitationalpotential: ξ ( r ± ) = (cid:104) Φ( r )Φ( r ) (cid:105) ± . (3.36)Once again, the ± labels correspond to the off-center LTB and Bianchi models, respectively.Likewise, the ± notation in (cid:10) a (cid:96) m a ∗ (cid:96) m (cid:11) ± indicates that each covariance matrix correspondsto one of each correlation function in (3.32). Usually, deviations from isotropy and homo-geneity are quantified directly in terms of the power spectrum ± P ( k ) = ˆ d r ± e − i k · r ± ξ ( r ± ) . (3.37)– 13 –or example, by expanding ± P ( k ) in harmonics one can show that [7, 10] (cid:104) a (cid:96) m a (cid:96) m (cid:105) ± = i (cid:96) ± (cid:96) π (cid:88) (cid:96),m ˆ k d k ± P (cid:96)m ( k ) j (cid:96) ( k ∆ η ) j (cid:96) ( k ∆ η )( − m G (cid:96)(cid:96) (cid:96) − m,m m , (3.38)where G (cid:96) (cid:96) (cid:96) m m m = ˆ d ˆ n Y (cid:96) m (ˆ n ) Y (cid:96) m (ˆ n ) Y (cid:96) m (ˆ n ) (3.39)are the Gaunt coefficients (see Appendix A.3) and ± P (cid:96)m ( k ) are the multipolar coefficients ofthe power spectrum. Then, from (3.38) and the coupling properties of the Gaunt coefficients,a feature in the power spectrum can be directly converted into a feature in the covariancematrix. In fact it is easy to extract ± P (cid:96)m ( k ) from the coefficients in Table 1 by means of theso-called Hankel transform (see Appendix A.1): ± P (cid:96)m = 4 πi − (cid:96) ˆ ∞ r ± d r ± j (cid:96) ( kr ± ) ξ (cid:96)m ( r ± ) . (3.40)However, it is interesting to have an expression for the covariance matrix directly in terms of ξ (cid:96)m . This can be obtained by inserting the above expression into (3.38), which gives (cid:104) a (cid:96) m a (cid:96) m (cid:105) ± = 89 (cid:88) (cid:96) ,m ˆ η r ± d r ± ξ ∗ (cid:96) m ( r ± ) J ( ± ) (cid:96) (cid:96) (cid:96) ( r ± ) G (cid:96) (cid:96) (cid:96) m m m , (3.41)where the coefficients J ( ± ) (cid:96) (cid:96) (cid:96) are implicitly defined in terms of the following integral J ( ± ) (cid:96) (cid:96) (cid:96) ( R, r , r ) ≡ i (cid:96) ± (cid:96) − (cid:96) ˆ ∞ k d kj (cid:96) ( kr ) j (cid:96) ( kr ) j (cid:96) ( kR ) . (3.42)This integral can be analytically solved and the result can be expressed in terms of Wigner 6-Jsymbols [35]. Parity symmetries of the Wigner 6-J symbols then result in several propertiesof the coefficients J ( ± ) (cid:96) (cid:96) (cid:96) [35]. For our discussion, the most relevant property is that thesecoefficients vanish whenever R lies outside the range | r − r | ≤ R ≤ r + r . (3.43)In the context of CMB, r = r = ∆ η and r ± = √ η √ ± ˆ n · ˆ n , so that ≤ r ± ≤ η . (3.44)This explains why the domain of the integral in (3.41) is limited. This result makes perfectsense since it is impossible to consider points in the CMB sphere whose separation is largerthan η .Returning to Table 1 we see that each geometry leaves its own fingerprint on the temper-ature spectrum. To the lowest order in β ii in this formalism both Bianchi-I and VII modelsproduce quadrupolar anisotropies whereas only the latter will alter the isotropic temperaturespectrum (i.e., the C (cid:96) s). Higher multipoles come from higher order corrections in β ii , but wewill not consider those here. Parity symmetry of these two models prevent even-odd couplingsof the harmonic coefficients [36]. For an off-center LTB universe, on the other hand, therewill be dipolar couplings as well as a change of the angular spectrum at low (cid:96) s, which depends– 14 –n the derivatives of the function ξ ( r − , r + ) evaluated at r + = 0 . Although one can obtainthese features by inserting the coefficients ξ (cid:96)m directly in (3.41), it is easier to relate them tothe Bipolar Spherical Harmonics (BipoSH) coefficients [9, 37] (see the Appendix A.4) ( ± ) A LM(cid:96) (cid:96) = 89 ˆ r ± d r ± ξ LM ( r ± ) J ( ± ) (cid:96) (cid:96) L ( r ± ) F (cid:96) (cid:96) L , (3.45)where the set of coefficients F (cid:96) (cid:96) L were defined in (A.13), and relate these to the covariancematrix using (A.11). Thus, Bianchi-I and VII geometries lead to a quadrupolar BipoSH A M(cid:96) (cid:96) , whereas an off-center LTB produces a dipolar BipoSH A M(cid:96) (cid:96) .As a final remark, we emphasize that expression (3.41) should be seen as containinganisotropies and inhomogeneities only from the initial conditions, after which we assume theuniverse to be pure FLRW. In particular, contributions resulting from integrated effects fromthe last scattering surface to us – like the effect induced by the lensing potential in anisotropic[18, 38] and inhomogeneous [27, 39] universes – cannot be extracted from this formalism inits present form. On the other hand, the multipolar features resulting from the coefficients inTable 1 would still be preserved – perhaps in an integrated version – as long as perturbationsare functions of the background coordinates. Indeed this is corroborated by the results of[18, 38], where quadrupolar corrections in the correlation of weak-lensing convergence of large-scale structure in a Bianchi-I spacetime was found. Furthermore, since (2.8) was designedto work on scalar functions, it cannot be directly applied to the cross-correlations betweenscalar, vector and tensor perturbations, which are known to couple dynamically through theevolution of the background shear [40, 41]. Nevertheless, since tensor fields are still seen asexternal fields in a fixed background, tensor correlators should be expected to obey a similarformalism as the one presented here (see also [1]). We postpone a deeper investigation ofthese issues to a future work. It is straightforward to extend this formalism to non-Gaussian correlation functions. Let ϕ be any N -point ( N > correlation function. Repeating the arguments leading to condition(2.8) then gives N (cid:88) j =1 K µ a ∂ µ ϕ | j = 0 . (4.1)The first non-trivial non-Gaussian statistical moment is the three-point correlation function(3pcf). Let us consider this function in an FLRW universe, where there are both translationaland rotational symmetries. Imposing invariance under the vector T µx = (1 , , gives thefollowing condition on ϕ : ∂ϕ∂x + ∂ϕ∂x + ∂ϕ∂x = 0 . (4.2)This is solved by any ϕ with an arbitrary dependence on the variable X defined as X = lx + mx + nx , l + m + n = 0 , (4.3)with constants ( l, m, n ) . However, the constraint on these constants allows us to write X = l ( x − x ) + m ( x − x ) , (4.4)– 15 –hich shows that ϕ can actually depend on the two “base” combinations ( x − x ) and ( x − x ) . Since we have no more constraints, these are the simplest combinations of x -coordinates on which ϕ can depend. Applying the same reasoning for translations along y -and z -directions then gives ϕ homog. ( r , r , r ) = ϕ homog. ( r − r , r − r ) (4.5)which is the most general homogeneous three-point function [7, 42]. To obtain an expressionwhich is also invariant under rotations we introduce u = r − r and v = r − r and simplynote that the task of finding ϕ ( u , v ) invariant under rotations has already been solved in§3.1. The solution is simply a function depending on the modulus of u ± v (see eqs. (3.8)).In terms of the original variables this becomes ϕ F L = ϕ F L ( | r − r | , | r + r − r | ) . (4.6)Since the reasoning we used to arrive at this result might not be entirely obvious, we notethat rotations around the z -axis of the vectors r , r and r are equal to rotations aroundthe z -axis of u and v : R z = (cid:18) x ∂∂y − y ∂∂x (cid:19) + (cid:18) x ∂∂y − y ∂∂x (cid:19) + (cid:18) x ∂∂y − y ∂∂x (cid:19) = x ∂∂u y − y ∂∂u x + x ∂∂v y − y ∂∂v x + x (cid:18) − ∂∂u y − ∂∂v y (cid:19) − y (cid:18) − ∂∂u x − ∂∂v x (cid:19) = (cid:18) u x ∂∂u y − u y ∂∂u x (cid:19) + (cid:18) v x ∂∂v y − v y ∂∂v x (cid:19) , where we have introduced a (hopefully obvious) new notation for the components of u and v . An equivalent result holds for R y and R x , as one can easily check. Then, by repeatingthe analysis of §3.1 we find ϕ = ϕ ( | u − v | , | u + v | ) which gives (4.6) upon replacing u and v by their definitions.There is one interesting remark we would like to make about eq. (4.6). Notice that if wemake the identification r = w the 3pcf will have exactly the same functional dependence asthe 2pcf in eq. (3.7) – namely, a Gaussian correlation in an universe with a special point. Thissuggests that the bispectrum (the Fourier transform of the 3pcf) in a FLRW universe couldmimic the power spectrum in a off-center LTB universe. Interestingly, it has been argued thata (statistically homogeneous and isotropic) bispectrum in the strong squeezed limit will inducestatistical anisotropies in the power spectrum [43]. In Fourier space the power spectrum andbispectrum have the form (assuming statistical homogeneity and isotropy) ξ ( k , k ) = P ( k ) δ ( k + k ) , (4.7) ϕ ( k , k , k ) = B ( k , k , k · k ) δ ( k + k + k ) . (4.8)In the squeezed limit k ≈ − k the wave vector k ≈ corresponds to a long wavelengthperturbation which is equivalent to a spatial gradient. This gradient modulates the lowerorder statistics leading to an effective power spectrum which is now anisotropic: P ( k ) → P eff ( k ) . We add to these the fact that the delta δ ( k + k + k ) in the bispectrum breaks Note that since ϕ ( | u − v | , | u + v | ) is equivalent to ϕ ( u, v, u · v ) , eq. (4.6) is also equivalent to ϕ ( | r − r | , | r − r | , ( r − r ) · ( r − r )) , which appears to be more common in the literature [42]. – 16 –he statistical independence previously existing between k and k in (4.7). Thus, in thepresence of a bispectrum ϕ ( k , k , k ) , ξ ( k , k ) is no longer translational invariant . In realspace, the similarity between (4.6) and (3.7) is just reflecting the fact that the third pointin the 3pcf could itself be seen as a “special” point. Analogously, a special point of an LTBuniverse will itself correlate with any two points previously correlated.As one last application let us consider the 3pcf in a LTB universe. Rotational invariancearound R µz = ( − y, x, gives (cid:18) x ∂ϕ∂y + x ∂ϕ∂y + x ∂ϕ∂y (cid:19) − (cid:18) y ∂ϕ∂x + y ∂ϕ∂x + y ∂ϕ∂x (cid:19) = 0 . (4.9)We could try solving this equation with the introduction of two new variables X = lx + mx + nx and Y = ly + my + ny . This would give X ∂ϕ∂Y − Y ∂ϕ∂X = 0 . (4.10)The use of characteristics would then tell us that ˙ X = − Y and ˙ Y = X , which implies that ϕ is a function of the constant combination X + Y . This solution however is not the mostgeneral one. To see that, note that in the absence of translational invariance the constraintin (4.3) no longer holds. In this case we have l + m + n = 2 p (4.11)for some constant p . We can thus rewrite the variable X as X = l ( x − x ) + m ( x − x ) + p ( x + x ) + p ( x + x ) − p ( x + x ) (4.12)with an analogous expression for Y . This tell us that there are actually five “base” combina-tions on which ϕ will depend, i.e., ϕ = ϕ ( x − x , x − x , . . . , x + x , . . . ) . Repeating theanalysis for R y and R x , which we hope by now has become clear, we find ϕ = ϕ ( | r − r | , | r − r | , | r + r | , | r + r | , | r + r | ) . (4.13)Note in particular that the combination | r + r | cannot be neglected, as one could haveexpected from a naive comparison with (3.8). The reason is that while r − r is linearlydependent on r − r and r − r (thus eliminating the need to include the former), thevectors r + r , r + r and r + r are linearly independent (the plane made by any two ofthem will not contain the third), and thus should all be included.Finally, the 3pcf in an off-center LTB universe is ϕ w = ϕ w ( | r − r | , | r − r | , | r + r − w | , | r + r − w | , | r + r − w | ) . (4.14)This can be obtained from (4.13) as follows: since the location of the special point w isarbitrary, the 3pcf should satisfy a shift symmetry analogous to (3.9). We thus shift allpoints in (4.13) by an arbitrary amount a and w so as to make the result shift invariant. Thisgives the above result. Note that this holds for any k , k and k , regardless of the squeezed limit. – 17 – Final remarks
Correlation functions belong to the core of modern cosmology. The perspective of extend-ing the Λ CDM model to inhomogeneous, anisotropic, and non-Gaussian universes dependscrucially on our abilities to model and measure such functions with increasing levels of so-phistication. In this work we have introduced a novel formalism which allows us to fix thefunctional dependence of correlation functions given the underlying spacetime (continuous)symmetries. Given a set of Killing vectors, we have found a set of first order partial differen-tial equation which can be solved for the functional dependence of the correlation function.The method works for arbitrary N -point correlators as long as one stays in the Born approx-imation – that is, as long as cosmological perturbations can be treated as external fields ina fixed background. We have also provided a general solution to the two-point correlationfunction which naturally introduces the time dependence, provided one finds a set of triadvectors commuting the Killing vector fields. This solution is particularly useful in applicationsto Bianchi cosmologies, where such triad of vectors can always be found [28, 29].We have successfully applied the formalism to the two-point function in three differentcosmological spacetimes, namely, the anisotropic and spatially flat solutions of Bianchi typeI and VII , and to the case of an off-center LTB universe, which includes the standard LTBmodel as a special case. Specializing to the case of CMB temperature fluctuations, we haveprovided asymptotic expansions of these correlation functions around the known Friedman-nian case. Each spacetime leaves its own multipolar fingerprint on the CMB covariance matrix (cid:104) a (cid:96) m a (cid:96) m (cid:105) . To the lowest order in the expansion parameters, we have found that Bianchi-Ispacetimes lead to quadrupolar couplings (cid:104) a (cid:96) m a (cid:96) ± ,m (cid:105) while preserving the isotropic an-gular spectrum C (cid:96) . Bianchi VII models, on the other hand, lead to quadrupole couplings aswell as suppression of the C (cid:96) s, whereas an off-center LTB metric leads to dipolar couplingsand a modification of the C (cid:96) s – the latter depending on a free function which has to be fixedby solving the photon transport equations in this geometry.We have also applied the method to infer the functional dependence of (non-Gaussian)three point correlation functions to the (well-known) case of a FLRW universe, and also tothe case of an off-center LTB universe. As a byproduct we have found a formal link betweenthe three-point correlation function in an FLRW universe and a Gaussian 2pcf in an off-centerLTB universe. This link results from the fact that a universe with a strong dependence onthe three-point correlation function is geometrically degenerate to a Gaussian universe witha special point.We would like to end with some remarks on the limitations and possible extensions of theformalism. First we stress that, although the method can be used to quickly give the CMBmultipolar couplings in a given geometry, it cannot be expected to give more information thanthat. The case of Bianchi-I is a clear example. While the quadrupolar couplings we found hereare compatible with the result of more in-depth analysis, the present formalism cannot predictthe oscillations in the power spectrum resulting from linear perturbation theory [34, 40] northe correlation between scalar and tensor modes arising from the dynamical couplings withthe shear [15, 41]. Second, we have not considered the case of spin functions, which are ofcentral importance to the physics of polarization and weak-lensing of the CMB. The case ofvector two-point functions in de Sitter spacetimes have been addressed in [1] using a differentformalism, where it was found that it also has the same symmetries of the background space.In the present formalism this conclusion is not immediate since equation (2.8), when appliedto more general tensor correlators, will introduce new terms coming from the Lie derivative of– 18 –he tensor. We postpone such analysis to future publications. Nonetheless, we emphasize thatthe method developed here is general, and can be equally useful in applications to quantumfield theory in curved spacetime. Acknowledgments
This work was supported by Conselho Nacional de Desenvolvimento Tecnológico (CNPq)under grant 485577/2013-5. O.H.M thanks CAPES for financial support. We also thankCyril Pitrou for insightful remarks on the final version of this work.
A Miscellanea
We gather here some useful formulae and results which were used in the main text.
A.1 Power spectrum and Hankel transform
In the examples considered in this work, the correlation function lacks global rotation sym-metry, so that it depends on the vector connecting two points in the following manner ξ ( r , r ) = ξ ( r ± ) , r ± = r ± r . (A.1)In this case the power spectrum also becomes a direction-dependent function of the Fouriervector ± P ( k ) = ˆ d r ± e − i k · r ± ξ ( r ± ) . (A.2)To relate the multipolar coefficients of P ± to those of ξ we first use Rayleigh’s expansion e − i k · r ± = 4 π (cid:88) (cid:96),m i − (cid:96) j (cid:96) ( kr ± ) Y (cid:96)m (ˆ k ) Y ∗ (cid:96)m (ˆ n ± ) , r ± = r ± ˆ n ± . (A.3)Next we decompose both P ± and ξ into spherical harmonics and use their orthogonalityrelation to express the multipolar coefficients of each function. The result is the Hankeltransform of the power spectrum (see ref. [44] for its use in cosmology) ± P (cid:96)m = 4 πi − (cid:96) ˆ ∞ r ± d r ± j (cid:96) ( kr ± ) ξ (cid:96)m ( r ± ) . (A.4) A.2 Covariance matrix
Since expression (3.41) is not very popular, we show here that it does lead to the correctresults when the universe is homogeneous and isotropic. For a FLRW universe we have ξ (cid:96) m ( r − ) = ξ ( r − ) δ (cid:96) δ m . (A.5)For this multipolar combination the Gaunt factor becomes G (cid:96) (cid:96) m m = ( − m √ π δ (cid:96) (cid:96) δ m , − m . (A.6)– 19 –oreover J ( − ) (cid:96) (cid:96) ( r − ) = ˆ ∞ k d k j (cid:96) ( k ∆ η ) j (cid:96) ( k ∆ η ) j ( kr − )= π η ) r − ˆ ∞ d xJ (cid:96) +1 / ( x ) sin(2 ax ) , a ≡ r − / (2∆ η )= π η ) r − P (cid:96) (cid:0) − a (cid:1) where in the last step we have used integral 6.672.5 of ref. [45]. Next we recall that r − = 2 (∆ η ) (1 − cos γ ) = 4 (∆ η ) a (A.7)which gives J ( − ) (cid:96) (cid:96) ( r − ) = π η ) r − P (cid:96) (cos γ ) . (A.8)Bringing everything together in expression (3.41) we find (cid:104) a (cid:96) m a (cid:96) m (cid:105) − = 89 ˆ η r − d r − ξ ∗ ( r − ) J ( − ) (cid:96) (cid:96) ( r − ) ( − m √ π δ (cid:96) (cid:96) δ m , − m , = 2 π η ) ( − m √ π (cid:20) ˆ η r − d r − ξ ∗ ( r − ) P (cid:96) (cos γ ) (cid:21) δ (cid:96) (cid:96) δ m , − m , We now note that r − d r − = (∆ η ) d ( − cos γ ) so that (cid:104) a (cid:96) m a (cid:96) m (cid:105) = ( − m (cid:20) π ˆ − d (cos γ ) ξ F L ( γ ) P (cid:96) (cos γ ) (cid:21) δ (cid:96) (cid:96) δ m , − m , = ( − m C (cid:96) δ (cid:96) (cid:96) δ m , − m , where in the last line we have used eq. (2.22) and ξ F L ( γ ) = ξ / √ π . A.3 Gaunt coefficients
The Gaunt coefficients result from the integral of three spherical harmonics over the sphere.They are given by [46] G (cid:96) (cid:96) (cid:96) m m m = ˆ d ˆ n Y (cid:96) m (ˆ n ) Y (cid:96) m (ˆ n ) Y (cid:96) m (ˆ n )= (cid:114) (2 (cid:96) + 1) (2 (cid:96) + 1) (2 (cid:96) + 1)4 π (cid:18) (cid:96) (cid:96) (cid:96) (cid:19) (cid:18) (cid:96) (cid:96) (cid:96) m m m (cid:19) . where the × matrices are the Wigner 3-J symbols. The Gaunt coefficients are identicallyzero whenever the sum (cid:96) + (cid:96) + (cid:96) is an odd number, and whenever m + m + m (cid:54) = 0 . A.4 Bipolar power spectrum
The bipolar power spectrum [9, 37] are the harmonic coefficients of the correlation func-tion when expanded in a basis of bipolar spherical harmonics [47]. They are related to thecovariance matrix as A LM(cid:96) (cid:96) = (cid:88) m ,m (cid:104) a (cid:96) m a (cid:96) m (cid:105) ( − M + (cid:96) − (cid:96) √ L + 1 (cid:18) (cid:96) (cid:96) Lm m − M (cid:19) . (A.9)– 20 –sing the identity [46] (cid:88) L,M (2 L + 1) (cid:18) (cid:96) (cid:96) Lm m M (cid:19) (cid:18) (cid:96) (cid:96) Lm (cid:48) m (cid:48) M (cid:19) = δ m m (cid:48) δ m m (cid:48) , (A.10)the inverse relation is found to be (cid:10) a (cid:96) m a ∗ (cid:96) m (cid:11) = ( − m + (cid:96) − (cid:96) (cid:88) L,M √ L + 1 A LM(cid:96) (cid:96) (cid:18) (cid:96) (cid:96) Lm − m − M (cid:19) . (A.11)By inserting (3.41) into (A.9) and using [46] (cid:88) m ,m (cid:18) (cid:96) (cid:96) (cid:96) m m m (cid:19) (cid:18) (cid:96) (cid:96) Lm m − M (cid:19) = δ L(cid:96) δ M, − m √ L + 1 (A.12)one arrives at (3.45), where the coefficients F (cid:96) (cid:96) L were defined by F (cid:96) (cid:96) L = ( − (cid:96) − (cid:96) (cid:114) (2 (cid:96) + 1) (2 (cid:96) + 1) (2 L + 1)4 π (cid:18) (cid:96) (cid:96) L (cid:19) . (A.13) References [1] B. Allen and T. Jacobson,
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